the connectedness of symmetric and skew-symmetric degeneracy loci… · 2018-11-16 · symmetric...

transactions of theamerican mathematical societyVolume 313, Number 1, May 1989

THE CONNECTEDNESS OF SYMMETRIC AND SKEW-SYMMETRIC

DEGENERACY LOCI: EVEN RANKS

LORING w. tu

Abstract. A degeneracy locus is the set of points where a vector-bundle map

has rank at most a given integer. Such a set is symmetric or skew-symmetric

according as whether the vector-bundle map is symmetric or skew-symmetric.

We prove a connectedness result, first conjectured by Fulton and Lazarsfeld,

for skew-symmetric degeneracy loci and for symmetric degeneracy loci of even

ranks.

Introduction

In this paper we prove a conjecture of Fulton and Lazarsfeld on the con-

nectedness of symmetric and skew-symmetric degeneracy loci, when the rank

is even. Since skew-symmetric maps always have even ranks, this settles the

conjecture in the skew-symmetric case. The odd-rank symmetric case remains

open.

Let E and F be complex vector bundles over an irreducible variety X, and

u: E —» F a vector-bundle morphism between them. For a given nonnegative

integer r, the degeneracy locus of rank r of u is defined to be

(0.1) Dr(u) = {xGX\ranku(x)<r}.

The concept of a degeneracy locus is quite general, and includes as special

cases all projective hypersurfaces, complete intersections, zero sets of sections

of a vector bundle, and dependency loci of sections. However, in algebraic

geometry one often encounters not only general vector-bundle morphisms, but

morphisms satisfying symmetry conditions. If L is a line bundle over X, a

morphism u : E®E —* L is said to be symmetric if it is symmetric on each fiber;

similarly for a skew-symmetric morphism. Their degeneracy loci are defined as

in (0.1). As examples of symmetric bundle morphisms we cite (1) the second

fundamental form of a smooth hypersurface in a projective space, and (2) the

differential of the period map of a family of varieties. For more background

information on degeneracy loci the reader is referred to [T].

Received by the editors January 4, 1988.

1980 Mathematics Subject Classification (1985 Revision). Primary 14M12; Secondary 14C17,14F05.

Key words and phrases. Connectedness, determinantal varieties, degeneracy loci, symmetric de-

generacy loci, skew-symmetric degeneracy loci, ample vector bundles, isotropic subspaces, rank.

©1989 American Mathematical Society

0002-9947/89 $1.00 + 1.25 per page

381

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

382 L. W. TU

These three types of vector-bundle morphisms may be viewed as sections

of the bundles Hom(E,F), (Sym2E*) <g> L, and (/\2E*) <g> L respectively,

where E* denotes the dual of E. In [FL1] Fulton and Lazarsfeld proved

that if Hom(E ,F) is ample and « is a section of Hom(E ,F), then Dr(u) is

connected provided its expected dimension, dim X-(e-r)(f-r), is at least one.

They conjectured the following statements for symmetric and skew-symmetric

degeneracy loci [FL2, Remark (2), p. 50].

Conjecture 0.2. Let E be a vector bundle of rank e and L a line bundle over the

irreducible variety X. Suppose u: E ® E —► L is a symmetric bundle map and

r a nonnegative integer. If (Sym E*) <g> L is ample and dimA" - (e~2+1) > 1,

then Dr(u) is connected.

Conjecture 0.3. Suppose u: E <g>E —> L is a skew-symmetric bundle map and r

a nonnegative even integer. If (f\ E*) ® L is ample and dimX - (<?2r) > 1,

then Dr(u) is connected.

We prove both conjectures for even values of r, thus settling Conjecture 0.3

completely. The strategy of the proof is to represent a symmetric degeneracy

locus of even rank as the image of a zero locus on a Grassmann bundle, as in

Pragacz [P], using the characterization of the rank of a symmetric matrix by

the dimensions of its isotropic subspaces. It then suffices to prove the connect-

edness of the zero locus. If the zero locus were that of a section of an ample

vector bundle, its connectedness would follow immediately from the well-known

Lefschetz-type theorem for ample vector bundles [S]. Unfortunately, the rele-

vant vector bundle on the Grassmann bundle is no longer ample, and so we try to

prove the connectedness by computing cohomology groups. This is done using

a construction by which the cohomology of the complement of the zero locus is

related to the cohomology of an affine variety. One hitch here is that the crucial

cohomology comparison lemma of Fulton and Lazarsfeld [FL1, Lemma 1.6]

applies only to birational maps with very restrictive fiber dimensions and is not

applicable to our situation. We formulate and prove a generalization (Lemma

3.6) of the Fulton-Lazarsfeld lemma that compares the cohomology of a variety

M with that of an affine image of M, allowing arbitrary positive-dimensional

fibers and thereby completing the proof.

The author is grateful to Ron Donagi for helpful discussions, and to Rick

Miranda whose Mountain West Algebraic Geometry Workshop provided a fo-

rum for this work and whose comments greatly simplified the exposition of this

paper.

Notational convention. Throughout this paper, by dim X we mean the com-

plex dimension of X.

1. Isotropic subspaces

In this section we establish the characterization of the rank of a symmetric

map by the dimensions of its isotropic subspaces. The result is undoubtedly


SYMMETRIC AND SKEW-SYMMETRIC DEGENERACY LOCI 383

well-known to the experts and its inclusion is justified only by the lack of a

suitable reference. So let E be a complex vector space of dimension e, and let

E* be its dual Hom^C). A linear map f:EÊ* is said to be symmetric

if f = f*, where f*:EÊ* is the dual of /. Equivalently, a linear map

/:.£—► /?* is symmetric if and only if its associated bilinear form (pf: ExE —►

C is symmetric. Since there is a bijection between the set of all symmetric maps

/:£—►£■* and the set of all symmetric bilinear forms <p: E x E —> C, we will

speak interchangeably of the two sets. For example, we define the kernel of

a symmetric bilinear form to be the kernel of its associated linear map. The

following characterization of the rank of a linear map is a standard fact from

linear algebra.

Proposition 1.1. A linear map f:E-+E* has rank < r if and only if dim(ker /)

> e - r, where e = dim E.

Recall that an isotropic subspace of a symmetric bilinear form tp : E x E —> C

is a subspace V of E such that <p\v: V x V —> C is identically zero. For a

symmetric map f:EÊ* there is a more interesting characterization of the

rank in terms of isotropic subspaces (Proposition 1.4).

Proposition 1.2. Let 4>: E x E -^ C be a symmetric bilinear form on a vector

space E of dimension e. Then the dimension of any isotropic subspace of <p is

at most e - ((rank<p)f 2).

Proof. Denote by K the kernel of the associated linear map f:E^>E* and

let F be a maximal isotropic subspace. Clearly, by the symmetry of /, K c V

and <f> induces a bilinear map tp': (V/K) x (E/V) —► C. By the definition of

K, the associated linear map V/K —► (E/V)* of <f>' is injective. Therefore,

dim V - dimK < e - dim V.

Since dim K = e - rank (p, the proposition follows. D

A symmetric bilinear map <f>: E x E —► C is said to be nondegenerate if

<j)(x ,y) = 0 for all y in E implies that x = 0.

Proposition 1.3. Let <p: E x E —» C be a symmetric bilinear form on a vector

space E of dimension e. Then

(i) All maximal isotropic subspaces of tf> have the same dimension.

(ii) If rank(7J is even, say 2p, then a maximal isotropic subspace of <f> has

dimension e - p.

(iii) If rank tp is odd, say 2p + I, then a maximal isotropic subspace of <f>

has dimension e - p - 1.

Proof, (i) Denote by K the kernel of tp. Then </3 induces a nondegenerate

symmetric bilinear map <p' : E/K x E/K —► C. Any maximal isotropic subspace

V of <j> must contain K, and V/K is a maximal isotropic subspace of cf> . By

[L, Corollary 2, p. 362] all maximal isotropic subspaces of the nondegenerate

<j>' have the same dimension. So if Vx and V2 are two maximal isotropic


384 L. W. TU

subspaces of tp, then dim Vx/K = dim V2/K. It follows that Vx and V2 have

the same dimension.

(ii) Suppose rank <p = 2p . Relative to some basis of E, the bilinear form

<j> is represented by the matrix

p v

h °„ 0p p

is the p x p zero matrix. This

p , which is the max-

imum possible by Proposition 1.2.

(iii) Suppose rank(/> = 2p + I. Relative to some basis of E, the bilinear

form tp is represented by the matrix

-2p

where / is the p x p identity matrix and 0

shows that <p has an isotropic subspace of dimension e

1

0.

0„0„ -2p-l

which shows that <p has an isotropic subspace of dimension e — p — I, the

maximum possible by Proposition 1.2. D

Using this proposition we can construct a table of the dimensions of the

maximal isotropic subspaces F of a symmetric bilinear map <j> : E x E —> C.

rankcp 0 1 2

dim V e e - 1 e - 1

3 4 5 6 7 8e-2 e-2 e-3 e-3 e-4 e-4

Proposition 1.4. A symmetric bilinear map <p: E x E —» C has rank < 2p if and

only if it has an isotropic subspace of dimension e - p.

Proof. This follows directly from the table above. D

Remarks. (1) As Proposition 1.4 shows, the dimensions of the isotropic sub-

spaces characterize only the bilinear forms of ranks at most an even integer.

(2) Analogously, Proposition 1.4 remains true for a skew-symmetric bilinear

map.

2. A DUALITY CONSTRUCTION

Let E be a vector space with coordinates zx,... ,zg. We think of the dual

space E*, with coordinates ax, ... ,ae , as the space of all hyperplanes in E.

Then there is a classic duality:

{points in £} <—► {linear forms on E*) = T(P(E*),(?(l)),

z = z, >o z* = £ ziar

¡=i



Globalizing this simple observation, if now £ is a vector bundle over a

variety X, then a section i of £ over X induces a section s* of cf(l) over

the projective bundle P(E*), where s*(x, [e*]) = e*(s(x)) for e* G E*.

t?(l)

E P(E*)

X

Proposition 2.1. If E is a ranke vector bundle over a variety X and s is a

section of E over X, then P(E") - Z(s*) is a Cf~ -bundle over X - Z(s),

where Z( ) denotes the zero locus of a section.

Proof. We examine the zero locus of s* on P(E*) fiber by fiber. If s(x) =

0, then s* vanishes on the entire fiber P(£*)- At a point x in X where

s(x) ^ 0, the restriction of s* to the fiber P(E*) vanishes precisely at the point

(x, V), where V is the hyperplane in Ex defined by (s(.x))* = 0. Therefore,

P(E*) -Z(s*) fibers over X-Z(s) with fiber Ce~x.

Corollary 2.2. Under the hypothesis of the proposition,

Hq(P(E*) - Z(s*);Z) ~ Hq(X - Z(s);Z).

The vector bundle E over X is said to be ample if the tautological line

bundle <f(l) over P(E*) is ample. This implies immediately that P(E*) -

Z(s*) is an affine variety.

3. Symmetric degeneracy loci

Given a rank e vector bundle E and a line bundle L over a variety X

and a symmetric bundle map u: E ® E —> L over X, recall that the rank r

degeneracy locus of u is defined to be

Dr(u) = {x G X\ rank u(x) < r}.

For the sake of simplicity we assume in this section that X is smooth and L

is trivial. Our goal now is to prove the following theorem.

Theorem 3.1. Let u: E <g> E —» C be a symmetric bundle map over a smooth

variety X, and p a nonnegative integer. If Sym E* is ample and dim X —

(e~22p+x) > 1, then D2p(u) is connected.

By the correspondence between symmetric linear maps and quadratic forms,

u may be regarded as a section of the vector bundle Sym2/?* over X. Using the

characterization of the rank in Proposition 1.4 we can represent the degeneracy

locus D2 (u) as the image of a zero locus, as follows.

Let it : G = G(e - p ,E) —> X be the bundle whose fiber at each point x in

X is the Grassmannian G(e - p ,Ex) of (e -/z)-dimensional subspaces of the

fiber E , and let -S be the universal subbundle over G. The inclusion map


386 L. W. TU

5 «-+ n*E over G induces the restriction map Sym n*E* —» Sym 5*. The

section m: X —► Sym £* gives rise naturally to a section (n*u)(x, V) = u(x) of

7i* Sym E* over C7(e -p,E) and hence by restriction to a section of Sym2S*

over G :

t(x,V) = u(x)\v: V^V*.

n*Sym2E*

Z(t) cG=G(e-p,E)

■*- Symz5

Sym2£*

£>2» c x

Proposition 3.2. The degeneracy locus D2(u) is the image under n of the zero

locus Z(t).

Proof. This proposition follows from the following series of equivalences:

x G D2(u) iff rank u(x) < 2p

iff u(x) has an (e -/?)-dimensional isotropic subspace V c Ex

ifft(x,V) = 0

iff(x,V)GZ{t). D

Hence, it suffices to prove the connectedness of the zero locus Z(t).

Lemma 3.3. Let M be a connected compact orientable manifold of (real) dimen-

sion n and A a subset of M. Then

(i) A is nonempty iff Hn(M - A ; Z) = 0 ;

(ii) A is connected if Hn (M - A ; Z) = Hn~ ' (M - A ; Z) = 0.

Proof. Both statements are consequences of Lefschetz duality

H"~q(M -A) = Hq(M ,A)

and the long exact sequence of a pair:

^HX(M,A)->H0(A)^H0(M) HQ(M,A)-*0.

Z D

By this lemma, the connectedness of the locus Z(t) in the Grassmann bundle

G = G(e - p ,E) follows from the vanishing of the two cohomology groups



H2dimG(G-Z(t);Z) and H2dimG~x(G - Z(t);Z). Instead of computing the

cohomology of G — Z(t) directly, we can apply Corollary 2.2 and compute the

cohomology of P(Sym S) - Z(t*) first. The advantage here is that one can

map P(Sym2 S) - Z(t*) to the affine variety P(Sym2 E)-Z(u), and use the

Leray spectral sequence to compare the cohomology of these two spaces. As an

affine variety the cohomology of P(Sym E) -Z(u) vanishes above its middle

dimension. So the computation of H*(G-Z(t)) will be based on the following

diagram:

(3-4)

P - Z(t') cP = P(SymzS)

Isame

cohomology

r

G-Z(t)cG=G(e -p,E)

Sym2£* P(Sym2E) = P'DP'-Z(u*)

D2p(u) c X

Let P = P(Sym25) and P' = P(Sym2 E). In this diagram P - Z(t*) is

an affine-space bundle over G - Z(t) by Proposition 2.1 and therefore has the

same cohomology as G - Z(t). By hypothesis Sym E* is an ample vector

bundle over X, so tfp,(I) is an ample line bundle over P', and P' - Z(u)

is an affine variety. The map h : P(Sym S) —► P(Sym E) is induced from the

inclusion of Sym V in Sym2 Ex :

h(x, V c Ex,<Pg Sym2 V) = (x,4>G Sym2£v).

Proposition 3.5. The map h: P —> P' sends P - Z(t*) to P' - Z(u).

Proof. By the definition of t*,

t*(x ,VcEx,<Pg Sym' V) = (u(x)\v)* (<P) = u(x)*(<P)

= u*(x,tp g Sym Ex) = u(h(x,V,cp)).

Soif t*(x,V,<p) ¿ 0, then u(h(x, V ,<p)) ¿ 0 and h sends P - Z(t") toP'-Z(u). a

To compute H*(P — Z(t*)) we will apply the following cohomology lemma,

whose proof is postponed to the next section. Denote by N the set of natural

numbers, {0,1,2,3, ...}.


388 L. W. TU

Lemma 3.6. Let h: M -* Y be a surjective proper morphism from any variety M

to an affine variety Y. Suppose there is a strictly increasing function d: N —► N

and a sequence of closed subvarieties

■■■cYk+xGYkcYk_xc---cY0 = Y

such that for all x in Yk - Yk+X,

Define

d(k) = dimch (x).

R = max{dimr Y. + 2d(k)}.k>0 c k

Then Hq(M,Z) = 0 for all q>R.

To apply this lemma, let P(Dk(Sym2 E)) be the subvariety of P(Sym2£')

consisting of elements of rank at most k , and let

Uk = P(Dk(Sym2E)) n (P' - Z(u)).

The image of h: P - Z(t*) -* P' - Z(u) lies in Ue_p . Define Yk = Ue_p_k .

Then

-cYk+lcYkcYk_xc---cY0

and dimyfc = dimP' - (p+k2+ï). For (x,<pG Sym2Ex) GYk- Yk+X , <p may

be viewed as a symmetric map <p: E* —> Ex of rank exactly e - p - k , so that

im tp is an (e - p - k)-dimensional subspace of Ex. Thus,

h~](x,<p) ~ {V g G(e -p,Ex)\im<p c V c Ex}

~ G((e-p) -(e-p- k),Ex/imtp)

~G(k,p + k).

Therefore, for (x,tp) G Yk - Yk+X, dimch~x(x,(p) = pk. In the cohomology

lemma (3.6) set d(k) = pk . Then

R = max{dim Y. + 2pk\k>0 K

= maxZc>0

k>o { 2 J

For integer values of p and k, the minimum of (k - p + l)(k - p)/2 is 0,

when k = p or p - 1. Therefore,

R = dimP' -p = dimX+ (e*l]-l-p.

By Lemma 3.6,

(3.7) Hq(P-Z(t*);Z) = 0 for q > dimZ+ (e + l j -p.



A straightforward computation shows that

dim* > (e~22+l) + 1 <*2dimC7-l >dimX+ (e + M -p.

Therefore, if dim* > (i-2/+l) + 1, then

H2dimG(G- Z(t);Z) = H2dimG-x(G - Z(t);Z) = 0.

By Proposition 3.2 and the remark following Lemma 3.3, Z(t) and hence

D2p(u) is connected.

4. Proof of the cohomology lemma

We prove here Lemma 3.6, which allows one to compare the cohomology of

a variety M with the cohomology of an affine image Y of M. For technical

reasons that will be apparent shortly, we extend the definition of the fiber-

dimension function d to {-1,0,1,2,...} by defining d(-l) = -1 . This

makes sense, for the fibers over the empty set Y_X-YQ are of course the empty

sets. In the notations of Lemma 3.6, applying the Leray spectral sequence to the

surjective map h : M -* Y, we find that Hj(Y,R¡htZ) abuts to Hi+j(M,Z).

So it suffices to prove that Hj(Y,R'hJZ) = 0 for all i + j > R.

Claim. Whenever there is a k G N such that

(4.1) i > 2d(k - I) and j > dim Yk ,

then Hi(Y,RihtZ) = 0.

Proof. Suppose i > 2d(k - 1). For dimension reasons, R'htZ vanishes on

Y - Yk . Thus, R'htZ is concentrated on Yk and

Hj(Y,RihtZ) = Hj(Yk,RihtZ).

As a direct image of a constructible sheaf, R'htZ is again constructible on Yk .

By a standard theorem, the cohomology of a constructible sheaf on an affine

variety vanishes above the dimension of the variety. So HJ(Yk, R'htZ) = 0 for

j > dim Yk . This concludes the proof of the claim.

Since d: {-1,0,1,2, ...}—»{-1,0,1,2, ...} is a strictly increasing func-

tion, given any nonnegative integer z we can choose k such that

2d(k) > i > 2d(k - 1).

(This inequality is the reason the domain and the image of the function d must

contain -1, since i may be 0.) If i + j > R, then

j>R-i>R- 2d(k) > dim Yk ,

where the last inequality follows from the definition of R. By (4.1),

HJ(Yk , R'htZ) = 0 for all i + j> R.


390 L. W. TU

5. Generalizations

In §3 we proved the connectedness conjecture (0.2) for an even-rank sym-

metric degeneracy locus under the assumptions that the ambient space X is

smooth and that the line bundle L is trivial. It is not difficult to remove both

restrictions.

In the proof of §3 the smoothness assumption was needed only to invoke

Lefschetz duality and to conclude that if Hq(G-Z(t) ;Z) = 0 for q = 2dimC7,

2dim G - 1 , then H (G,Z(t);Z) = 0 for q = 0,1. Thus, to generalize from

a smooth variety to an arbitrary irreducible variety X, what is needed is a

substitute for Lefschetz duality. This is provided by an argument in [FL1],

which in fact goes through verbatim in the present situation. Their idea is first

to show that one may assume X normal. For if v : X —» X is the normalization

of the irreducible variety X, then X is also irreducible. Furthermore, the

section u of Sym2/?* over X pulls back to a section v*u of Sym2/?* over

X, and Dr(u) is the image of Dr(v*u) under v. Since the pullback of an

ample vector bundle under a finite morphism is again ample, v* Sym E* is

ample. So it suffices to prove the connectedness of Dr(v*u). Replacing X by

X, we may assume X normal. Then the Grassmann bundle G over X is also

normal, and the following lemma applies.

Lemma [FL1, Lemma 1.3]. Let G be a normal projective variety of dimension

m and let Z c G be a closed algebraic subset. Then there is an injection

HX(G,Z;Q)^H2m_x(G-Z;Z).

By the universal coefficient theorem, Hq(G - Z(t);Z) = 0 for q = 2 dim G,

2dimG - 1 implies that H2dimG_x(G - Z(t);Q) = 0 [M, Corollary 56.4 and

Theorem 53.5]. Hence, Hx(G,Z(t);Z) = 0 and Z(t) is connected. This

concludes the proof when X is an irreducible variety.

Finally, suppose L is an arbitrary line bundle. If L happens to be the square

of a line bundle, say L ~ K <g> K, then (Sym2 E*) <g> L ~ Sym2(£'* <g> K) and

Conjecture 0.2 for even ranks would follow immediately from Theorem 3.1

(but with X an irreducible instead of a smooth variety). The following lemma

of Bloch and Gieseker effectively allows one to assume that every bundle is a

square.

Lemma 5.1 [BG, Lemma 2.1]. Let L be a line bundle on a projective variety

X and d a positive integer. Then there exists a projective variety Y, a finite

surjective morphism f:Y—*X, and a line bundle K on Y such that f*L ~

Proof. See [BG, p. 114 or FL2, Lemma 1.1]. n

Returning to the proof of Conjecture 0.2 for even ranks, we take d = 2 in

Lemma 5.1. Then

/*((Sym2 E*) g L)) =¡ (/* Sym2 £*) ® K2 ~ Sym2(/*£* ® K).



The degeneracy locus Dr(f*u) in Y maps surjectively onto Dr(u) in X. Since

the pullback of an ample vector bundle under a finite morphism is again ample,

we conclude by Theorem 3.1 (again with X irreducible instead of smooth) that

Dr(f*u) is connected. Therefore, Dr(u) is also connected.

6. Skew-symmetric degeneracy loci

By the argument of §5, Conjecture 0.3 can also be reduced to the case where

X is a smooth variety and L is a trivial line bundle. To prove Conjecture2 2

0.3 under these assumptions, replace Sym by f\ in Diagram 3.4. This gives

a map h: P - Z(C) -* P* - Z(u), where P = P(A2S) and P1 = P(/\2E).

Let Uk = P(Dk(/\2 E)) n P' - Z(u). The proof proceeds as in the even-rank

symmetric case, except now because the rank of a skew-symmetric matrix is

always even, the image of h lies in U or Ue_p_x depending on whether

e - p is even or odd, and the rank drops by two at a time. To combine the two

cases in one proof, let

2k, if e - p is even,I = {

2k + 1, if e - p is odd,-{

and define Yk = Ue_p_¡. Then the image of h lies in YQ

dimYk = dimP'-(P + 1}

and

cYk+lcYkcYk_xc---cY0.

If (x, tp G/\2EX) G Yk-Yk+X, then

h~ (x,cp) ~ G(l,p + I) and dimch~ (x,tp) =pl.

In the cohomology lemma (3.6),

R=ma,{dimP'-^+2l)+2pl}

= maJdimP'+p-^-l+^-ink>0 { 2 }

= dim P' +p = dim X + (^) -l+p.

By Lemma (3.6),

Hq(P -Z(C) ;Z) = 0 forq>A\mX+ie}\+p.

Again one checks easily that

dimX> \}P ) + 1 <*2dimC7- 1 >dimX+ (e\ +P-

So under the hypotheses of Conjecture 0.3,

H2dimG(G-Z(t);Z) = H2dmG-x(G-Z(t);Z) = 0

and the connectedness of D2p(u) follows as before.


392 L. W. TU

References

[ACGH] E. Arbarello, M. Cornalba, P. Griffiths, and J. Harris, Geometry of algebraic curves,

Springer-Verlag, New York, 1985.

[BG] S. Bloch and D. Gieseker, The positivity of the Chern classes of an ample vector bundle, Invent.

Math. 12(1971), 112-117.

[FL1] W. Fulton and R. Lazarsfeld, On the connectedness of degeneracy loci and special divisors,

Acta Math. 146 (1981), 271-283.

[FL2] W. Fulton and R. Lazarsfeld, Positive polynomials for ample vector bundles, Ann. of Math.

118(1983), 35-60.

[HT] J. Harris and L. W. Tu, On symmetric and skew-symmetric determinantal varieties, Topology

23(1984), 71-84.

[HT2] J. Harris and L. W. Tu, The connectedness of symmetric degeneracy loci: odd ranks, Proc.

Minisemester on Commutative Algebra and Algebraic Geometry (Warsaw, 1988) (to appear).

[L] S. Lang, Algebra, Addison-Wesley, Menlo Park, Calif., 1984.

[M] J. R. Munkres, Elements of algebraic topology, Addison-Wesley, Menlo Park, Calif., 1984.

[P] P. Pragacz, Cycles of isotropic subspaces and formulas for symmetric degeneracy loci, Proc.

Minisemester on Commutative Algebra and Algebraic Geometry (Warsaw, 1988) (to appear).

[S] A. Sommese, Submanifolds of abelian varieties. Math. Ann. 233 (1978), 229-256.

[T] L. W. Tu, Degeneracy loci, Proc. Internat. Conf. Algebraic Geometry (Berlin, 1985), Teubner

Verlagsgesellschaft, Leipzig, 1986, pp. 296-305.

[T2] L. W. Tu, The connectedness of degeneracy loci, Proc. Minisemester on Commutative Algebra

and Algebraic Geometry (Warsaw, 1988) (to appear).

Department of Mathematics, Tufts University, Medford, Massachusetts 02155


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